If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional analysis can be defined as the practice of studying relations between physical quantities by recognising the dimensions of them. These dimensions are independent of numerical multiples and similar constants. The quantities present in the world can be described as a function of the fundamental dimensions.

Dimensional Formula

Dimensional formula of a quantity is the expression indicating the powers wherein the fundamental units are to be increased to obtain one unit of a derived quantity. For instance, if Q is an unit of a derived quantity which is defined by Q = M^aL^bT^c, then M^aL^bT^c is known as the dimensional formula with exponents a, b and, c that are called the dimensions.

Dimensional Formula for Resistance

The dimensional formula of resistance can be expressed as:

M¹L²T^-3I^-2

Here,

M = Mass
I = Current
L = Length
T = Time

Resistance (R) = Voltage × Current^-1… [1]

Now, because, Voltage (V) = Electric Field × Distance = [Force × Charge^-1] × Distance

Accordingly, dimensional formula of charge = current × time = I¹T¹

Dimensional formula of voltage = [Force × Charge^-1] × Distance

= [M¹ L¹ T^-2] × [I¹T¹]^-1 × [L¹]

= [M¹ L² T^-3I^-1] … [2]

On replacing equation (ii) in equation (i) we get,

Resistance (R) = Voltage × Current^-1

R = [M¹ L² T^-3I^-1] × [I]^-1 = [M¹ L² T^-3I^-2]

Hence, resistance is dimensionally depicted as M L² T^-3I^-2.

Dimensional Formula of Capacitance

The dimensional formula of Capacitance is expressed by:

M^-1L^-2T⁴I²

Here,

M = Mass
I = Current
L = Length
T = Time

Derivation of Capacitance

Capacitance (C) = Charge × Voltage^-1. . . [1]

Since, Charge = Current × Time

∴ Therefore, the dimensional formula of charge can be give as = [I¹T¹] . . . . [2]

And, Voltage = Electric Field × Distance . . [3]

Now, as per Electric Field,

Electric Field = [Force × Charge^-1]

Hence, dimensional formula of force and charge equals [M¹ L¹ T^-2] and [I¹T¹] respectively.

∴ The dimensional formula of Electric Field = [M¹ L¹ T^-2] × [I¹T¹]^-1

= [M¹ L¹ T^-3 I^-1] . . . [4]

On replacing equation [4] in equation [3] we can obtain,

The dimensional formula of Voltage = [M¹ L¹ T^-3 I^-1] × [L¹]

= [M¹ L² T^-3I^-1] . . . [5]

On replacing equations [5] and [2] in equation [1] we can obtain,

Hence, Capacitance = Charge × Voltage^-1

Or, as can be expressed, C = [I¹T¹] × [M¹ L² T^-3I^-1]^-1 = [M^-1 L^-2 T⁴I²]

Thus, the Capacitance can be dimensionally expressed as [M^-1 L^-2 T⁴I²].

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If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional Formula

Dimensional Formula for Resistance

Dimensional Formula of Capacitance

Derivation of Capacitance

Related Questions

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SUBSCRIBE TO OUR NEWS LETTER

If R, C and L are fundamental quantities in a circuit like resistance, capacitance and inductance in W, then find dimensional formula for resistance and capacitance.

Dimensional Formula

Dimensional Formula for Resistance

Dimensional Formula of Capacitance

Derivation of Capacitance

Related Questions

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3. If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

4. Draw the number of scattered particles versus the scattering angle graph for scattering of alpha particles by a thin foil. Write two important conclusions that can be drawn from this plot.

5.Four independent waves are expressed as \[ (i)\; y_1=A_1\sin\omega t, \] \[ (ii)\; y_2=A_2\sin 2\omega t, \] \[ (iii)\; y_3=A_3\cos\omega t, \] \[ (iv)\; y_4=A_4\sin\left(\omega t+\frac{\pi}{3}\right) \] The interference between two of these waves is possible in

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